Internet pioneer Marc Andreessen is doubling down on bitcoin amid turbulence in the virtual-currency world, in a bet that widespread adoption of the currency will fuel the growth of new businesses and technologies.

Venture-capital firm Andreessen Horowitz, where Mr. Andreessen is a co-founder and partner, has made about $50 million of investments in the area—believed to be more than any other firm—from a $1.5 billion fund, the firm...